Amplification of electromagnetic fields by a rotating body

In 1971, Zel’dovich predicted the amplification of electromagnetic (EM) waves scattered by a rotating metallic cylinder, gaining mechanical rotational energy from the body. This phenomenon was believed to be unobservable with electromagnetic fields due to technological difficulties in meeting the condition of amplification that is, the cylinder must rotate faster than the frequency of the incoming radiation. Here, we measure the amplification of an electromagnetic field, generated by a toroid LC-circuit, scattered by an aluminium cylinder spinning in the toroid gap. We show that when the Zel’dovich condition is met, the resistance induced by the cylinder becomes negative implying amplification of the incoming EM fields. These results reveal the connection between the concept of induction generators and the physics of this fundamental physics effect and open new prospects towards testing the Zel’dovich mechanism in the quantum regime, as well as related quantum friction effects.

where I is the current in the coil wrapped around the toroid.The geometrical factor β = (0.40 ± 0.03) T/A has been measured experimentally.The vector b 0 = (1, 0, 0) T e iωt represents the linear polarization of the field oscillating at frequency ω.To our purposes it is convenient to write b 0 as the sum of two rotating vectors, co-rotating and counter-rotating with respect to the rotor, such that: In order to see the amplification effect we are looking for, calculations are carried out in the reference frame co-rotating with the rotor.In this frame, we write the vector b 0 as the sum of a corotating (-) and a counter-rotating (+) terms: We define ω ± = ω ± Ω as the corotating (ω − ) and counter-rotating (ω + ) frequencies of the field with respect to the rotor, with ω the frequency of the field B 0 and Ω the rotation frequency of the rotor in the lab frame.From now on we drop the prime for simplicity, since all calculations are carried out in the rotating frame.
Our purpose is to find the form of the vector potential A, which allows us to compute the magnetic dipole moment induced in the rotor, and the magnetic flux Φ = β|m 0 | back-reflected by the magnetic dipole into the toroid circuit.
To this aim, we solve Maxwell's equations, which describe the interaction of the magnetic field B generated by the circuit and the conductive sphere and we find the equation for the vector potential [2]: where σ is the electrical conductivity and µ 0 and µ r are the vacuum and relative permeability, respectively.In order to solve this last equation for A, we divide the space in two regions, inside and outside the sphere with radius a, and consider the following three boundary conditions at the interface between the two regions and at infinity: where B = µ 0 µ r H and r is the position vector in the co-rotating reference frame centered at the center of the sphere.The magnetic field outside the sphere B out = B 0 + B refl is composed of two components: the incident magnetic field generated by the circuit B 0 plus the magnetic field reflected from the rotor B refl .Let us remind that B 0 is composed by a co-rotating and a counter-rotating component (see eq. ( S3)).The magnetic field inside the rotor B in is given by the component of the incident field transmitted in the scattering.
Considering the last boundary condition, the complex vector potential A for a uniform applied field B 0 can be generally written as A = 1 2 [F (r)B 0 ] × r [2].We can generalize the vector potential A to our case as a sum of co-rotating and counter-rotating contributions: The function F (ω, r), such that F (ω, r → ∞) = 1, can be computed by substituting eq.(S8) in eq.(S4), finding [2]: where c (ω ± ) = √ −iµ 0 µ r σω ± .The functions f (x) and D(x) are given by: where These last equations define the explicit form of the vector potential A = A − + A + in the whole space.
It is now possible to compute the magnetic dipole moment m 0 induced in the rotating sphere, which will be composed by two components: one co-rotating with the sphere and another one counter-rotating.From the relation between m 0 and the reflected component of the vector potential we find It is worth noting that in order to correctly calculate m 0 , we have subtracted a factor 1 from F (ω ± , a) corresponding to the contribution of the incident field B 0 .
In the last equation we have defined the susceptibility : i.e., the complex response function of the sphere to a rotating magnetic field.Note that the susceptibility can be decomposed as χ = χ ′ − iχ ′′ , where the real part χ ′ represents the conservative component in-phase with the applied field while the imaginary part χ ′′ is out-of-phase and therefore dissipative.
Looking at the amplification from the point of view of the LC circuit, the rotating magnetic moment will couple a magnetic flux Φ into the circuit with peak value Φ = β|m 0 |.We can thus write a relation where the flux response function is related to the sphere susceptibility by α (ω) = β 2 χ (ω).Since the flux in the laboratory frame is oscillating at ω, the associated voltage induced in the circuit is then: ), so that we can define the resistance R and the inductance L generated by the presence of the rotor as: The factor α ′ gives the inductance L generated by the presence of the rotor in the gap, while ωα ′′ can be seen as a resistance R induced by the rotor into the circuit.Hence amplification corresponds to a negative resistance that leads to power emission into the electromagnetic (EM) mode as opposed to the expected (for a non-rotating or slowly rotating rotor) power absorbed from the EM mode.We observe that the resistance R is composed by a co-rotating and a counter-rotating components.We have amplification only if the co-rotating one is larger than the counter-rotating one.This is possible since in general α (ω − Ω) ̸ = α (ω + Ω).
To understand this point it is instructive to look more in detail at the behaviour of the response function χ(ω) of the cylinder with respect to the EM field.The response of the system is characterized by a length scale δ(ω) = (σµ 0 µ r ω) −1/2 , which in our model is encoded in the coefficient c(ω) = √ i/δ(ω), and is usually known as the penetration depth.The magnitude of the dissipative component of the response function |χ ′′ | (see Fig. S1) is maximized when the penetration depth approaches the system size (characterised by the sphere radius a).This means that there is a maximum absorption (positive χ ′′ ) and a maximum amplification (negative χ ′′ ) when ω − is such that the penetration depth is of the order of the sphere radius a (we are referring to the corotating frequency ω − being the only one to undergo negative Doppler-shift when Ω > ω, in the present experiment).Likewise, we can define a scaling frequency ω * = (µ 0 µ r σa 2 ) −1 at which the penetration depth equals the radius a of the sphere.
The dissipative component χ ′′ as a function of the normalized frequency ω − /ω * for a purely conductive sphere with µ r = 1 is plotted in Fig. S1.The function is odd, flipping sign when crossing 0 towards negative frequencies, which is at the core of the Zel'dovich-like amplification observed in this work.It is worth making two considerations here.First of all noticing that the dissipative response function χ ′′ behavior has to be considered in the corotating reference frame.Secondly we are considering it only for the frequency ω − since it is the only one achieving negative frequencies in this system when Ω > ω.However, the total dissipative response function has two contributions, χ ′′ (ω − ) + χ ′′ (ω + ), as shown in Eq. (S20) (and Eq. ( 4) in the main manuscript).Furthermore, in Fig. S1 we observe that χ ′′ achieves a maximum absolute value at ω m = 11.6ω* (ω * = 2π × 8.4 Hz in this system) and drops to 0 in both limits |ω| ≪ ω m and |ω| ≫ ω m .For |ω| ≪ ω m the dissipation vanishes because no eddy currents are induced in the sphere, while for |ω| ≫ ω m eddy currents are confined to a vanishingly thin skin layer δ ≪ a.
When the sphere is rotating at Ω and the field is a superposition of co-rotating and counter-rotating fields, as discussed above, we have two dissipative components, shown in Figure S2, as function of the normalized rotation frequency for two representative field frequencies ω = 50ω * and ω = 500ω * .The corotating term becomes negative when Ω > ω i.e. when the corotating frequency ω − = ω − Ω < 0. The sum of the two terms becomes negative for Ω larger than a certain critical frequency Ω c > ω.The relative width of of the transition becomes narrower at higher ω, however its absolute value is set by ω m , so it is independent of ω.

THE CYLINDER MODEL
The rotating body used for the experiment being a metallic cylinder, we also compute the amplification of a cylindrical body: we will compare this model with the sphere one and the experimental data at the end of this section.For symmetry reasons, analytical solutions in the cylindrical case exist only in the limit of infinite length.Let us consider a non rotating with infinite length in a transverse oscillating magnetic field, along x.To compute the effect of the cylinder we write the oscillating magnetic field back-reflected from the cylinder as a vector B = (B r , B ϕ , B z ) in cylindrical coordinates.Considering the geometry of our system the B z component is null since the external magnetic field is transverse to the cylinder and no induced ẑ component is anticipated.This leaves us with a 2D problem [4].Solving Maxwell equations for this problem gives the components of the back-reflected field from the cylinder: Figure S3 shows a comparison between the experimental results shown in Fig. 3 of the main paper, with both the sphere and the infinite cylinder model.Clearly, the sphere model captures the experimental behaviour more accurately.We attribute this experimental finding to the fact that the assumption of uniform applied field is more explicitly broken in the case of the infinite cylinder, due to the fall-off of the field at the edge of the gap.This leads to an overestimation of the rotor-induced effect.Conversely, in the sphere approximation the contributions at the edge of the gap are intrinsically suppressed by the spherical geometry.

SQUIRREL-CAGE INDUCTION GENERATOR VS SOLID ROTOR
Induction machines are a type of rotating machine where a stator circuit -the non-rotating part, usually embedded in a hollow cylinder -has alternating currents oscillating at a single frequency ω ac -the frequency of the power supply, which are used to generate a rotating field.The stator can have multiple magnetic poles p, hence the frequency of rotation of the stator magnetic field is ω s = ω ac /p.Secondary circuits are mounted on the rotor, placed inside the hollow stator cylinder, and rotate at frequency ω m -the mechanical rotation.The mechanism is based on the principle of induction: the rotor circuits are short-circuited and their currents are induced by the stator rotating field.
In most induction machines, a squirrel cage rotor is used.This is made of a solid cylinder of iron or steel carved by conductor bars, usually made of copper or aluminium, that are short-circuited by a conductive ring around the cylinder bases (see an sketch of the conductive bars in the typical squirrel cage circuit in Figure S4).The thickness and the shape of these conductive bars determines the inductive and resistive properties of the rotor circuits.In our analysis we consider the induction system described in [5] and we write the voltage V s induced by the stator on the rotor as where s = (ω s − ω m )/ω s is the so called slip, that can be related to the Zel'dovich condition considering the correspondence p → ℓ (ℓ = 1 in our case), ω m → Ω and ω ac → ω.M is the mutual inductance, while R rot , L rot are the resistance and inductance of the rotor's circuits.It is possible to find the effective susceptibility χ eff of the squirrel cage induction generator by recalling that α = β 2 χ eff .The geometrical factor β for the squirrel cage circuit is just the mutual inductance M .After some manipulation, we find: where the first term on the right hand side is the inductive component (χ ′ eff ) and the second is the resistive component (χ ′′ eff ), which indeed flips sign if the slip frequency (ω − Ω) < 0, i.e. the Zel'dovich condition is fulfilled.

EXPERIMENTAL APPARATUS
Fig. S5 shows photos of the experimental apparatus used to perform the experiment.The magnetic field is generated by a coil wound around a gapped toroidal ferrite core with square section 4 × 4 cm.An aluminum cylinder with radius a = 2 cm, mounted on a brush-less motor is fixed into the 4.4 cm gap in the ferrite core.As the cylinder can spin up to several hundred Hertz about its symmetry axis, protective clear plastic is mounted around it.The coil has 2 × 10 4 turns of 0.2 mm diameter copper wire, and is connected in series, which create a resonant circuit with the coils and improve the accuracy of measurements.
FIG. S5.The experimental apparatus used to perform the experiment.The cylindrical rotor is mounted into the gap in the toroid core, and can be driven to spin around its symmetry axis by a motor and controller.The cylinder can be removed for calibration measurements.Capacitors are connected in series with the coil windings.

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FIG. S1.Dissipative response of the sphere χ ′′ , normalized by the factor 2πa 3 /µ0 as a function of the normalized field frequency in the corotating frame ω−.
FIG. S2. (a): Dissipative response of the sphere χ ′′ , normalized by the factor 2πa 3 /µ0, as function of normalized rotation frequency for the the two components corotating (blue line) and counterrotating (cyan line).The red line is the sum of the two terms, showing that the total dissipation can indeed become negative.Here the field frequency is set to ω = 50ω * (dashed vertical line).Bottom panel: the same but now for ω = 500ω * (dashed vertical line).